# Properties

 Label 8624.2.a.bw Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$68.8629867032$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1078) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{5} - q^{9}+O(q^{10})$$ q + b * q^3 + b * q^5 - q^9 $$q + \beta q^{3} + \beta q^{5} - q^{9} + q^{11} + 2 \beta q^{13} + 2 q^{15} - 4 \beta q^{17} - 2 \beta q^{19} + 2 q^{23} - 3 q^{25} - 4 \beta q^{27} - 6 q^{29} + \beta q^{31} + \beta q^{33} - 2 q^{37} + 4 q^{39} + 4 \beta q^{41} - 4 q^{43} - \beta q^{45} - 3 \beta q^{47} - 8 q^{51} - 12 q^{53} + \beta q^{55} - 4 q^{57} - 3 \beta q^{59} - 4 \beta q^{61} + 4 q^{65} + 2 q^{67} + 2 \beta q^{69} + 10 q^{71} - 10 \beta q^{73} - 3 \beta q^{75} + 8 q^{79} - 5 q^{81} + 2 \beta q^{83} - 8 q^{85} - 6 \beta q^{87} - 5 \beta q^{89} + 2 q^{93} - 4 q^{95} - 11 \beta q^{97} - q^{99} +O(q^{100})$$ q + b * q^3 + b * q^5 - q^9 + q^11 + 2*b * q^13 + 2 * q^15 - 4*b * q^17 - 2*b * q^19 + 2 * q^23 - 3 * q^25 - 4*b * q^27 - 6 * q^29 + b * q^31 + b * q^33 - 2 * q^37 + 4 * q^39 + 4*b * q^41 - 4 * q^43 - b * q^45 - 3*b * q^47 - 8 * q^51 - 12 * q^53 + b * q^55 - 4 * q^57 - 3*b * q^59 - 4*b * q^61 + 4 * q^65 + 2 * q^67 + 2*b * q^69 + 10 * q^71 - 10*b * q^73 - 3*b * q^75 + 8 * q^79 - 5 * q^81 + 2*b * q^83 - 8 * q^85 - 6*b * q^87 - 5*b * q^89 + 2 * q^93 - 4 * q^95 - 11*b * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{11} + 4 q^{15} + 4 q^{23} - 6 q^{25} - 12 q^{29} - 4 q^{37} + 8 q^{39} - 8 q^{43} - 16 q^{51} - 24 q^{53} - 8 q^{57} + 8 q^{65} + 4 q^{67} + 20 q^{71} + 16 q^{79} - 10 q^{81} - 16 q^{85} + 4 q^{93} - 8 q^{95} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^11 + 4 * q^15 + 4 * q^23 - 6 * q^25 - 12 * q^29 - 4 * q^37 + 8 * q^39 - 8 * q^43 - 16 * q^51 - 24 * q^53 - 8 * q^57 + 8 * q^65 + 4 * q^67 + 20 * q^71 + 16 * q^79 - 10 * q^81 - 16 * q^85 + 4 * q^93 - 8 * q^95 - 2 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.41421 0 −1.41421 0 0 0 −1.00000 0
1.2 0 1.41421 0 1.41421 0 0 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8624.2.a.bw 2
4.b odd 2 1 1078.2.a.q 2
7.b odd 2 1 inner 8624.2.a.bw 2
12.b even 2 1 9702.2.a.dq 2
28.d even 2 1 1078.2.a.q 2
28.f even 6 2 1078.2.e.s 4
28.g odd 6 2 1078.2.e.s 4
84.h odd 2 1 9702.2.a.dq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.q 2 4.b odd 2 1
1078.2.a.q 2 28.d even 2 1
1078.2.e.s 4 28.f even 6 2
1078.2.e.s 4 28.g odd 6 2
8624.2.a.bw 2 1.a even 1 1 trivial
8624.2.a.bw 2 7.b odd 2 1 inner
9702.2.a.dq 2 12.b even 2 1
9702.2.a.dq 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{13}^{2} - 8$$ T13^2 - 8 $$T_{17}^{2} - 32$$ T17^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} - 32$$
$19$ $$T^{2} - 8$$
$23$ $$(T - 2)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 2$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 32$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 18$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} - 18$$
$61$ $$T^{2} - 32$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 200$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 8$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 242$$